Problem: Define
\[A = \frac{1}{1^2} + \frac{1}{5^2} - \frac{1}{7^2} - \frac{1}{11^2} + \frac{1}{13^2} + \frac{1}{17^2} - \dotsb,\]which omits all terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3, and
\[B = \frac{1}{3^2} - \frac{1}{9^2} + \frac{1}{15^2} - \frac{1}{21^2} + \frac{1}{27^2} - \frac{1}{33^2} + \dotsb,\]which includes only terms of the form $\frac{1}{n^2}$ where $n$ is an odd multiple of 3.

Determine $\frac{A}{B}.$
Explanation: We can start by taking out a factor of $\frac{1}{9}$ out of each term in $B$:
\[B = \frac{1}{9} \left( \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \frac{1}{9^2} - \frac{1}{11^2} + \dotsb \right).\]Note that we obtain all the terms in $A,$ so
\[B = \frac{1}{9} A + \frac{1}{9} \left( -\frac{1}{3^2} + \frac{1}{9^2} - \frac{1}{15^2} + \frac{1}{21^2} - \dotsb \right) = \frac{1}{9} A + \frac{1}{9} (-B).\]Then $9B = A - B,$ so $A = 10B.$  Therefore, $\frac{A}{B} = \boxed{10}.$